tbsv(3F)

NAME
     STBSV, DTBSV, CTBSV, ZTBSV - Solves a real or complex triangular
     banded system of equations

SYNOPSIS
     Real

        CALL STBSV (uplo, trans, diag, n, k, a, lda, x, incx)

     Double precision

        CALL DTBSV (uplo, trans, diag, n, k, a, lda, x, incx)

     Complex

        CALL CTBSV (uplo, trans, diag, n, k, a, lda, x, incx)

     Double complex

        CALL ZTBSV (uplo, trans, diag, n, k, a, lda, x, incx)

DESCRIPTION
     STBSV and DTBSV solve a real triangular banded system of equations.

     CTBSV and ZTBSV solve a complex triangular banded system of equations.

     These routines solve one of the following systems of equations, using
     the operation associated with each:

          Equations      Operation

          Ax=b x <- A-1x

          ATx=b x <- A-Tx

          AHx=b x <-     A-Hx (CTBSV and ZTBSV only)

     where

     * b and x are n-element vectors

     * A is either a unit or nonunit n-by-n upper or lower triangular band
       matrix with (k+1) diagonals

     * A-1 is the inverse of A

     * AT is the transpose of A

     * A-T is the inverse of AT


     * AH is the conjugate transpose of A

     * A-H is the inverse of AH


     On input, the right-hand side vector b is stored in the array argument
     x.  On output, the solution vector x overwrites b in the same array
     argument x.

     These routines have the following arguments:

     uplo      Character*1.  (input)
               Specifies whether the matrix is an upper or lower triangular
               matrix, as follows:

               uplo = 'U' or 'u': A is an upper triangular matrix.
               uplo = 'L' or 'l': A is a lower triangular matrix.

     trans     Character *1.  (input)
               Specifies the operation to be performed, as follows:

               trans = 'N' or 'n':  x <- A-1x

               trans = 'T' or 't': x <- A-Tx

               trans = 'C' or 'c': x <- A-Tx (STBSV, DTBSV), or

               x <- A-Hx (CTBSV, ZTBSV)

     diag      Character *1.  (input)
               Specifies whether A is unit triangular, as follows:

               diag = 'U' or 'u':  A is assumed to be unit triangular.
               diag = 'N' or 'n':  A is not assumed to be unit triangular.

     n         Integer.  (input)
               Specifies the order of matrix A.  n >= 0.

     k         Integer.  (input)
               uplo = 'U' or 'u':  k specifies the number of superdiagonals
               of matrix A.
               uplo = 'L' or 'l':  k specifies the number of subdiagonals
               of matrix A.
               k >= 0.

     a         Array of dimension (lda,n).  (input)
               STBSV: Real array.
               DTBSV: Double precision array.
               CTBSV: Complex array.
               ZTBSV: Double complex array.

               Before entry with uplo = 'U' or 'u', the leading (k+1)-by-n
               upper triangular part of array a must contain the upper
               triangular band part of the matrix of coefficients, supplied
               column-by-column, with the leading diagonal of the matrix in
               row (k+1) of the array, the first superdiagonal starting at
               position 2 in row k, and so on.  The top left k-by-k
               triangle of array a is not referenced.

               Before entry with uplo = 'L' or 'l', the leading (k+1)-by-n
               part of array a must contain the lower triangular band part
               of the matrix of coefficients, supplied column-by-column,
               with the leading diagonal of the matrix in row 1 of the
               array, the first subdiagonal starting at position 1 in row
               2, and so on.  The bottom right k-by-k triangle of array a
               is not referenced.

               When diag = 'U' or 'u', these routines assume that all
               elements of array a that represent diagonal elements of the
               matrix A are 1.  In this case, neither of these routines
               will reference any of the diagonal elements.

     lda       Integer.  (input)
               Specifies the first dimension of a as declared in the
               calling program.  lda >= (k+1).

     x         Array of dimension 1+(n-1) * |incx|.  (input and output)
               STBSV: Real array.
               DTBSV: Double precision array.
               CTBSV: Complex array.
               ZTBSV: Double complex array.
               Contains the vector x.

               On input, x contains the right-hand side vector b.  On
               output, the solution vector overwrites array x.

     incx      Integer.  (input)
               Specifies the increment for the elements of x.  incx must
               not be 0.

NOTES
     The following program segment transfers an upper triangular band
     matrix from conventional full matrix storage to band storage:

              DO 20, J = 1, N
                 M = K + 1 - J
                 DO 10, I = MAX( 1, J - K ), J
                    A( M + I, J ) = MATRIX( I, J )
          10     CONTINUE
          20  CONTINUE

     The following program segment transfers a lower triangular band matrix
     from conventional full matrix storage to band storage:

              DO 20, J = 1, N
                 M = 1 - J
                 DO 10, I = J, MIN( N, J + K )
                    A( M + I, J ) = MATRIX( I, J )
          10     CONTINUE
          20  CONTINUE

     Tests for singularity or near-singularity are not included in these
     routines.  You must perform such tests before calling these routines.

     These routines are Level 2 Basic Linear Algebra Subprograms (Level 2
     BLAS).

     When working backward (incx < 0), each routine starts at the end of
     the vector and moves backward, as follows:

          x(1-incx * (n-1)), x(1-incx * (n-2)), ..., x(1)